Graphics Reference
In-Depth Information
1
2
3
2
2
3
Ê
Á
ˆ
˜
(
)
q
=+
2
n
,
where
n
=
i
+
j
+
k
,
is the polar form of
q
.
We see that the polar form (20.4) of a quaternion is nothing startling but a useful
rewrite that associates an angle with a quaternion. However, there are other similar-
ities with the complex numbers. In fact, if we denote a
quaternion q
by the purely
formal
exponential notation
= re
q
n
q
(20.5)
where r, qŒ
R
and the pure quaternion
n
with
n
2
=-1 is defined by the polar form
(20.4) for
q
, then
20.3.4
Proposition.
(1) (re
q
n
)(se
h
n
) = rs e
(q+h)
n
(2) The inverse of e
q
n
is e
-q
n
.
(3) The De Moivre Theorem holds, that is,
m
(
)
+
(
)
cos
q
+
sin
q
n
=
cos
mm
q
sin
q
n
.
Proof.
This is straightforward and left as Exercise 20.3.1.
Finally, from the polar form (20.4) we also see that the complex numbers can be
imbedded in
H
in many ways, namely, we can use the imbedding
C
Æ
+Æ+
H
ab
i
ab
n
20.3.5
Proposition.
Let
q
be any nonzero quaternion and
z
any pure quaternion.
(1)
qzq
-1
is a pure quaternion.
(2) The map
3
3
s
q
:
RR
Æ
defined by
()
=
-1
s
q
zqzq
is an isometry of
R
3
. In fact, it is a rotation through an angle 2q about the oriented
line
L
through the origin with direction vector
n
, where
(
)
qq
=
cos
q
+
sin
q
n
is the polar form of
q
.
